Transverse momentum fluctuations as a probe of thermalization and collective dynamics of QGP

Rupam Samanta; Tribhuban Parida

REVIEW 4 major objections 5 minor 10 references

Transverse momentum fluctuations as a probe of thermalization and collective dynamics of QGP

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Steep drop in transverse momentum fluctuation variance signals quark-gluon plasma thermalization.

desk verdict A compact proceedings summary of earlier work, with a plausible but oversold thermalization interpretation: the sharp variance drop is produced by a fitted correlation, not an independent prediction. read the letter →

arxiv 2505.02706 v1 pith:2SYUVTGX submitted 2025-05-05 nucl-th nucl-ex

Rupam Samanta , Tribhuban Parida This is my paper · ORCID

classification nucl-thnucl-ex

keywords heavy-ioncollisiontransversemomentumfluctuationsquark-gluonplasmathermalizationcollectivedynamicsradialflowv0(pT)ultracentralcollisions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that event-by-event fluctuations of the mean transverse momentum per particle ([pT]) in ultracentral lead-lead collisions carry a direct, momentum-based signature of quark-gluon plasma thermalization. In the ultracentral limit the variance of [pT] at fixed charged-particle multiplicity drops sharply, and the paper attributes that drop to the rapid disappearance of impact-parameter fluctuations, which is only expected if the produced medium has thermalized and evolved hydrodynamically. The paper also introduces v0(pT), the correlation between the pT spectrum and [pT], as a differential map of these fluctuations, and argues that v0(pT)/v0 behaves like a true radial-flow observable, in the same way anisotropic flow probes collective dynamics. A sympathetic reader should care because the observable depends only on particle momenta, not azimuthal directions, giving an independent handle on QGP formation.

What carries the argument

The engine is a five-parameter two-dimensional correlated Gaussian distribution P([pT], Nch|b) for the joint event-by-event behavior of [pT] and charged multiplicity at fixed impact parameter, together with the variance decomposition in Eq. (1): Var(pT|Nch) equals the intrinsic variance at fixed b plus the variance of the mean [pT] over the impact-parameter distribution at fixed Nch. This decomposition turns the data curve into two physically separated contributions and isolates the b-fluctuation part that collapses in ultracentral collisions. The Pearson correlation r(b) between [pT] and Nch at fixed b is the load-bearing parameter that distinguishes a thermalized hydrodynamic medium from a nonthermal one. The differential observable v0(pT) is defined as the correlation of the normalized pT spectrum with [pT] fluctuations, normalized by N0(pT)σpT, and its scaled form v0(pT)/v0 is centrality independent.

What would settle it

Take the same ultracentral Pb+Pb events and compute the third and fourth cumulants of [pT] at fixed Nch; the Gaussian ansatz predicts they are negligible, so sizable values would mean the variance decomposition in Eq. (1) is contaminated and the sharp drop could have a different origin. Equally decisive, running a nonthermal event generator with the same multiplicity selection should not produce the steep ultracentral variance fall if the paper's thermalization mechanism is correct.

Watch

Extended reading notes

Core claim

The central claim is that the steep decrease of Var([pT]) with multiplicity in ultracentral Pb+Pb collisions is caused by the rapid decline of impact-parameter fluctuations at fixed multiplicity, and this decline is a natural consequence of the thermalization of the QGP medium. In the paper's model, the variance at fixed Nch has two parts: an intrinsic fluctuation at fixed impact parameter and a contribution from impact-parameter fluctuations at fixed Nch; in ultracentral events the second term vanishes, producing the observed sharp fall. A large Pearson correlation r(b) between [pT] and Nch at fixed b, present in hydrodynamic simulations and absent in a nonthermal model, is identified as the microscopic origin of the effect. The companion observable v0(pT), built from the correlation between normalized spectra and [pT] fluctuations, is shown to be centrality-independent when scaled by v0, to have flow-like mass ordering for identified particles, and to enable acceptance corrections that reproduce measured variances in different pT windows.

Load-bearing premise

The conclusion collapses if the joint distribution of [pT] and Nch at fixed impact parameter is not a two-dimensional correlated Gaussian with a single correlation coefficient r(b), because every extraction of the impact-parameter contribution and the thermalization signature passes through that ansatz.

Editorial extensions

If this is right

If the central claim is right, the ultracentral fall in Var([pT]) is a direct, momentum-based thermalization probe that does not rely on azimuthal anisotropies.
The scaled observable v0(pT)/v0 can serve as a centrality-independent differential radial-flow observable, with mass ordering for identified particles similar to elliptic flow.
Acceptance corrections derived from v0(pT)/v0 allow prediction of σpT in arbitrary pT windows from one reference measurement; the paper reproduces measured variances in two other pT windows this way.
Shear viscosity has negligible effect on v0(pT)/v0, while bulk viscosity produces small effects at high pT when the observable is plotted against pT/⟨pT⟩.
The same logic can be extended to small systems such as p+Pb and p+p as a test of whether QGP-like collectivity and thermalization appear there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A sharper test than the variance shape would be to measure r(b) directly by binning events in narrow Nch intervals at fixed centrality and correlating [pT] with Nch; the paper's mechanism implies this correlation should be large in ultracentral data and near zero in nonthermal models.
The Gaussian ansatz for P([pT], Nch|b) predicts that higher cumulants of [pT] at fixed Nch are negligible; measuring skewness or kurtosis in existing data would either confirm the ansatz or show where the thermalization attribution needs refinement.
If v0(pT) is truly radial flow, identified-particle v0(pT) should show a mass hierarchy that shifts with centrality in the same way v2 does, a check that could be done with already published spectra.
The mechanism also suggests that in smaller collision systems the ultracentral variance fall should be weaker or absent, providing a qualitative discriminator for whether a tiny droplet thermalizes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read this as a conference-proceedings condensation of the authors' own prior papers plus an interpretive claim. The most honest summary is: Eq. 1 is an exact law-of-total-variance decomposition; the model reproduces the ATLAS variance shape; the v0(pT) construction gives a differential radial-flow-like observable with a real consistency check; but the "direct evidence of thermalization" language goes beyond what the analysis establishes.

The v0(pT)/v0 acceptance-correction result is the part I trust most. Using one pT window as input and reproducing the variance in two other windows is a nontrivial, falsifiable check that does not depend on the thermalization interpretation. The left panel of Fig. 2 also shows useful insensitivity to viscosity details. That deserves credit.

The soft spot is exactly where the stress-test note points. In the correlated-Gaussian model, the impact-parameter term in Eq. 1 is generated by the Pearson correlation r(b). The sharp ultracentral drop therefore comes from the fitted value r_Nch = 0.676 combined with the Gaussian ansatz, fitted to the same ATLAS data. Calling that "direct evidence of thermalization" is circular unless r(b) is shown to emerge from an event-by-event hydrodynamic prediction rather than being fit. No uncertainties on r or the other parameters are reported, no goodness-of-fit is given, and the HIJING baseline appears only as a qualitative scatter plot. These omissions matter because the whole interpretive weight sits on r.

I do not think this is a fatal flaw. The thermalization interpretation is plausible and consistent with the hydro picture; the problem is that the paper does not yet distinguish a fitted description from a demonstrated mechanism. The heavy reliance on refs. [3] and [6] for the model details is fine for a proceedings note, but it makes the genuinely new content here thin. The citation pattern itself is appropriate: self-citations point to the papers where the model was actually derived.

Who gets value from this? Someone working on event-by-event fluctuations in heavy-ion collisions who wants a quick map of the argument and a reminder of the v0(pT) formalism. It is not a complete research paper, and the central claim should be treated as conditional until the authors provide parameter uncertainties, a quantitative non-thermal baseline, and preferably a multi-b hydrodynamic validation of the Gaussian ansatz.

My recommendation: send it to peer review rather than desk-reject, but require those additions if it is to carry the "direct evidence" claim. The v0(pT) piece alone merits referee time, and the fluctuation-vs-thermalization question is worth settling properly.

Referee Report

4 major / 5 minor

Summary. The manuscript proposes that the steep decrease of the variance of the event-by-event mean transverse momentum per particle, [pT], observed by ATLAS in ultracentral Pb+Pb collisions is caused by the rapid suppression of impact-parameter fluctuations at fixed multiplicity, and interprets this as direct evidence of QGP thermalization. It also introduces the differential observable v0(pT), defined as the correlation between the pT-spectrum and [pT] fluctuations, and argues that v0(pT)/v0 is centrality independent, shows a mass hierarchy, and can be used to extrapolate the multiplicity-dependent variance of [pT] between pT windows. The analysis is based on a 2D correlated Gaussian model for P([pT], Nch|b) with five parameters, fitted to the ATLAS variance data, and on hydrodynamic simulations at fixed impact parameter.

Significance. The paper's starting point, Eq. (1), is an exact law-of-total-variance decomposition, and the model reproduces the shape of the ATLAS variance data. If the thermalization interpretation is correct, the observable provides a momentum-magnitude-based signature of QGP thermalization complementary to azimuthal anisotropic flow, and v0(pT) is a potentially useful differential radial-flow observable. A strength is the explicit comparison with a non-thermal HIJING baseline and the use of hydrodynamic simulations. However, the central claim hinges on the fitted correlation coefficient r(b), and the current manuscript does not establish that r(b) is predicted rather than adjusted, nor does it provide uncertainties or goodness-of-fit measures; the HIJING comparison is only qualitative. The v0(pT) extrapolation in Fig. 2 is presented without quantitative errors.

major comments (4)

[Section 2, Eq. (1), Fig. 1] The central conclusion that the sharp variance drop is caused by impact-parameter fluctuations and constitutes direct evidence of QGP thermalization rests on the 2D correlated Gaussian ansatz for P([pT], Nch|b), with the Pearson correlation r(b) treated as a free parameter. In this model, the b-fluctuation term in Eq. (1) is equal to Var_b( r(b) [sigma_pT(b)/sigma_Nch(b)] [Nch - Nch(b)] | Nch ), so it vanishes identically if r(b)=0. The fitted value r_Nch=0.676 is adjusted to reproduce the same ATLAS data that are then interpreted as evidence for thermalization, making the inference circular unless r(b) is independently predicted. I request a validation in which r(b) (or the full conditional distribution) is obtained from event-by-event hydrodynamic simulations at several b values without fitting to the variance data, together with a quantitative goodness-of-fit estimate (e.g., chi^2 per degree of freedom) and uncertainties on the fitted parameters.
[Section 2, Fig. 1 (left)] The comparison with the non-thermal HIJING model is only qualitative. The paper states that HIJING lacks the strong correlation between [pT] and Nch, but it reports no quantitative correlation coefficient for HIJING, no uncertainty, and no resulting prediction for Var(pT|Nch) from HIJING. Without these numbers, the claim that the large r is missing in the non-thermal model and therefore signals thermalization is not established. Please provide the Pearson correlation for HIJING at fixed b and the HIJING-based variance prediction overlaid on Fig. 1 (right).
[Section 3, Fig. 2 (right)] The reproduction of the ATLAS c_k (variance) data for the lower pT windows using the acceptance correction factor C_A is not quantified. No residuals, chi^2, or uncertainties are reported, so it is unclear how accurately the model reproduces the data. Please provide a quantitative comparison and specify how C_A is computed from v0(pT)/v0 for each pT window, including the propagation of uncertainties from the fitted parameters.
[Section 2] The model parameters (Nch(b), sigma_Nch(b), pT(b), sigma_pT(b), r(b)) are not fully specified in the text; in particular, the b-dependence of each parameter and whether r is assumed constant (as the single quoted r_Nch suggests) are not stated. Since the conclusion depends on the b-dependence of these parameters, please present the explicit parameterizations used in the fit, or refer precisely to the equations in Ref. [3] where they are defined.

minor comments (5)

[Introduction] Typo: 'visocous' should be 'viscous' in the first sentence.
[Fig. 1] The label 'pT(0) = 9.357 MeV/c' is unphysical for the mean [pT]; if this value is actually sigma_pT(0), please relabel it.
[Fig. 1 (right)] The legend does not explicitly specify which curve corresponds to 'Intrinsic' and which to 'b fluctuation'; please add clear labels.
[Eq. (2)] The definition of v0(pT) uses delta N(pT) and delta pT without specifying whether averages are over all events or events in a fixed centrality/multiplicity bin; please define the averaging convention.
[Section 3, Fig. 2 (left)] The statement that v0(pT)/v0 is independent of centrality (or b) is supported by only a few simulation points; please quantify the variation or give a numerical range.

Circularity Check

2 steps flagged · score 6.0 of 10

Variance drop is generated by the fitted Gaussian correlation, so the 'direct evidence' claim is partly circular.

fitted input called prediction [Section 2, Eq. (1), Fig. 1; Section 4]
"The variance Var(pT|Nch) calculated from Eq. 1 is fitted to ATLAS data for 20 % most central events [3], shown in Fig. 1. ... Our model returns a very large value of r implying a very strong correlations between Nch and [pT] (missing in non-thermal model e.g. HIJING cf. Fig. 1) stemming as a direct consequence of the thermalization of the QGP medium."

In Eq. (1), the impact-parameter term is generated by the conditional mean of [pT] at fixed Nch. In the assumed 2D Gaussian, that conditional mean is proportional to the fitted Pearson correlation r(b). The same ATLAS variance data are used to fix r (e.g., r_Nch = 0.676), and the resulting term is then identified as the cause of the sharp ultracentral drop and as 'direct evidence' of thermalization. The claim is therefore a property of the fitted model, not an independent prediction; the drop is built in by construction.
ansatz smuggled in via citation [Section 2, paragraph on the 2D Gaussian ansatz and footnote 1]
"We assume that this distribution can be described by a 2D correlated Gaussian distribution P([pT], Nch|b) which characterized by five parameters: Nch(b), σNch(b), pT(b), σpT(b) and r(b), where r(b) denotes the Pearson correlation coefficient between [pT] and Nch at fixed b [3]."

The functional form that makes Eq. (1)'s second term nonzero is imported from Ref. [3], a previous paper by the same authors, rather than derived or independently tested here; the footnote sends the reader to [3] for details. Thus the 'direct evidence for thermalization' rests on an ansatz whose quantitative b-dependence is supplied by self-citation. This would be acceptable if the ansatz were validated against event-by-event hydro over the full b range, but the paper only shows a b=0 scatter plot and a qualitative HIJING comparison.

full rationale

The paper's central claim—that the sharp ultracentral fall of Var(pT|Nch) is caused by the rapid decline of impact-parameter fluctuations and is 'direct evidence' of QGP thermalization—is not an independent prediction. Eq. (1) decomposes the variance into an intrinsic part and an impact-parameter part, but in the assumed 2D correlated Gaussian the impact-parameter part is controlled by the Pearson coefficient r(b), which is among the parameters fitted to the ATLAS variance data. The fitted value r_Nch=0.676 is then interpreted as a thermalization signature, so the explanation is a restatement of the fit rather than a first-principles result. The b=0 hydro simulation and the HIJING scatter plot provide some independent qualitative support for a positive correlation in a thermalized medium, but no quantitative prediction of r(b) is made and tested across b. The v0(pT) acceptance correction in Sec. 3 is explicitly a consistency check ('reproduced') rather than a fresh prediction, so it does not add a circular step. The reliance on the authors' earlier Ref. [3] for the Gaussian ansatz and its parameters makes the inference additionally dependent on prior self-citation. Overall, the central prediction is partially circular: the observable that is claimed to diagnose thermalization is the same observable used to fix the model. Score 6.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim relies on a fitted Gaussian ansatz and a thermodynamic density-energy relation; no new physical entities are introduced. The fitted correlation r carries most of the thermalization evidence.

free parameters (3)

r (Pearson correlation between [pT] and Nch at fixed b) = 0.676
Fitted to ATLAS data and used as the main evidence of strong Nch-[pT] correlation and thermalization.
pT(0) (mean [pT] at b=0) = 9.357 MeV/c
Baseline normalization for the [pT]-Nch relation; fitted to ATLAS variance data.
alpha (scaling exponent in the Nch or [pT] relation) = 1.191
Scaling parameter whose value is fitted to data; no prior derivation is given.

assumptions (3)

ad hoc to paper The joint distribution P([pT], Nch|b) is a 2D correlated Gaussian with five parameters.
Section 2: 'We assume that this distribution can be described by a 2D correlated Gaussian distribution...' This is not derived; the entire variance decomposition and fit depend on it.
domain assumption Relativistic viscous hydrodynamics describes QGP dynamics and thermalization.
Introduction states that QGP dynamics can be described by relativistic viscous hydrodynamics; the model uses hydro simulations and attributes thermalization to hydro behavior.
domain assumption Larger particle density implies larger energy per particle, hence larger [pT].
Section 2: 'From relativistic thermodynamics, a larger density results in larger energy per particle eventually producing larger [pT].' This converts impact-parameter variation into [pT] variation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Transverse momentum fluctuations as a probe of thermalization and collective dynamics of QGP." pith.science (2026). https://pith.science/paper/2SYUVTGX

@misc{pith2026250502706,
  author       = {Pith},
  title        = {Pith review of: Transverse momentum fluctuations as a probe of thermalization and collective dynamics of QGP},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SYUVTGX}},
  note         = {Machine review of arXiv:2505.02706}
}

abstract

We study fluctuations of mean transverse momentum per particle ($[p_T]$) in ultrarelativistic heavy-ion collisions. We show that the steep fall in the variance of transverse momentum fluctuation in ultracentral Pb+Pb collision serves as a natural consequence of the thermalization of the QGP medium. We study the correlation between the spectra and $[p_T]$ which maps these fluctuations differentially and dubbed as $v_0(p_T)$. We highlight the importance of $v_0(p_T)$ showing that it plays similar role as anisotropic flow when probing the collective nature of QGP.

Figures

Figures reproduced from arXiv: 2505.02706 by the authors.

**Figure 2.** Left: Normalized charged particle spectra (top) and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 7 linked inside Pith

[3]

Samanta, S

R. Samanta, S. Bhatta, J. Jia, M. Luzum, and J.-Y . Ollitrault, Phys. Rev. C109, L051902 (2024), arXiv:2303.15323 [nucl-th]

arXiv 2024
[6]

Parida, R

T. Parida, R. Samanta, and J.-Y . Ollitrault, Phys. Lett. B 857, 138985 (2024), arXiv:2407.17313 [nucl-th]

arXiv 2024
[1]

Heinz and R

U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. 63, 123 (2013), arXiv:1301.2826 [nucl-th]

arXiv 2013
[2]

Busza, K

W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph]

arXiv 2018
[4]

Samanta, J

R. Samanta, J. a. P. Picchetti, M. Luzum, and J.-Y . Ollitrault, Phys. Rev. C108, 024908 (2023), arXiv:2306.09294 [nucl-th]

arXiv 2023
[5]

Schenke, C

B. Schenke, C. Shen, and D. Teaney, Phys. Rev. C 102, 034905 (2020), arXiv:2004.00690 [nucl-th]

arXiv 2020
[7]

Parida, QM25 talk at Frankfurt, https://indico.cern.ch/event/1334113/contributions/6289775/ attachments/3047185/5384425/QM2025_Tribhuban_Parida.pdf

T. Parida, QM25 talk at Frankfurt, https://indico.cern.ch/event/1334113/contributions/6289775/ attachments/3047185/5384425/QM2025_Tribhuban_Parida.pdf

arXiv
[8]

Acharya et al

S. Acharya et al. (ALICE), (2025), arXiv:2504.04796 [nucl-ex]

arXiv 2025

Show all 10 references

[9]

Aad et al

G. Aad et al. (ATLAS), (2025), arXiv:2503.24125 [nucl-ex]

2025
[10]

Bhatta, A

S. Bhatta, A. Dimri, and J. Jia, (2025), arXiv:2504.20008 [nucl-th] . 4

2025 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [3]

Samanta, S

R. Samanta, S. Bhatta, J. Jia, M. Luzum, and J.-Y . Ollitrault, Phys. Rev. C109, L051902 (2024), arXiv:2303.15323 [nucl-th]

arXiv 2024

[2] [6]

Parida, R

T. Parida, R. Samanta, and J.-Y . Ollitrault, Phys. Lett. B 857, 138985 (2024), arXiv:2407.17313 [nucl-th]

arXiv 2024

[3] [1]

Heinz and R

U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. 63, 123 (2013), arXiv:1301.2826 [nucl-th]

arXiv 2013

[4] [2]

Busza, K

W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph]

arXiv 2018

[5] [4]

Samanta, J

R. Samanta, J. a. P. Picchetti, M. Luzum, and J.-Y . Ollitrault, Phys. Rev. C108, 024908 (2023), arXiv:2306.09294 [nucl-th]

arXiv 2023

[6] [5]

Schenke, C

B. Schenke, C. Shen, and D. Teaney, Phys. Rev. C 102, 034905 (2020), arXiv:2004.00690 [nucl-th]

arXiv 2020

[7] [7]

Parida, QM25 talk at Frankfurt, https://indico.cern.ch/event/1334113/contributions/6289775/ attachments/3047185/5384425/QM2025_Tribhuban_Parida.pdf

T. Parida, QM25 talk at Frankfurt, https://indico.cern.ch/event/1334113/contributions/6289775/ attachments/3047185/5384425/QM2025_Tribhuban_Parida.pdf

arXiv

[8] [8]

Acharya et al

S. Acharya et al. (ALICE), (2025), arXiv:2504.04796 [nucl-ex]

arXiv 2025